Τελεστής Dirac
Τελεστής Dirac Operator, Dirac operator [[image:Numbers-03-goog.jpg|thumb|300px| Διακριτά Μαθηματικά Αριθμητική Αριθμοθεωρία Αριθμός Τελεστής ---- Αλγεβρικές Πράξεις Πρόσθεση Αφαίρεση Πολλαπλασιασμός Διαίρεση ---- Συνολοϊκές Πράξεις Συνολοϊκή Ένωση Συνολοϊκή Τομή ---- Λογικές Πράξεις Σύζευξη (Conjunction) Διάζευξη (Disjunction) Άρνηση (Negation) ---- Ιδιότητες Πράξεων Ανακλαστική Ιδιότητα Αντιμεταθετική Ιδιότητα Προσεταιριστική Ιδιότητα Επιμεριστική Ιδιότητα ]] - Ένα είδος τελεστών Ετυμολογία Η ονομασία "Τελεστής Dirac" σχετίζεται ετυμολογικά με το όνομα "Dirac". Εισαγωγή In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of first-order operators he introduced spinors. Ορισμός In general, let D'' be a first-order differential operator acting on a vector bundle ''V over a Riemannian manifold M''. If : D^2=\Delta, \, ::where ∆ is the Laplacian of ''V, then D'' is called a '''Dirac operator'. In high-energy physics, this requirement is often relaxed: only the second-order part of D''2 must equal the Laplacian. Examples '''Example 1:' D=-i ∂''x'' is a Dirac operator on the tangent bundle over a line. Example 2: We now consider a simple bundle of importance in physics: The configuration space of a particle with spin ½ confined to a plane, which is also the base manifold. It's represented by a wavefunction ψ: R'2 → '''C'2 :: \psi(x,y) = \begin{bmatrix}\chi(x,y) \\ \eta(x,y)\end{bmatrix} where x'' and ''y are the usual coordinate functions on 'R'2. χ specifies the probability amplitude for the particle to be in the spin-up state, and similarly for η. The so-called spin-Dirac operator can then be written :: D=-i\sigma_x\partial_x-i\sigma_y\partial_y,\, where σ''i'' are the Pauli matrices. Note that the anticommutation relations for the Pauli matrices make the proof of the above defining property trivial. Those relations define the notion of a Clifford algebra. Solutions to the Dirac equation for spinor fields are often called harmonic spinorshttp://eom.springer.de/S/s086780.htm. '''Example 3: The most famous Dirac operator describes the propagation of a free fermion in three dimensions and is elegantly written :: D=\gamma^\mu\partial_\mu\ \equiv \partial\!\!\!/, using the Feynman slash notation. Example 4: There is also the Dirac operator arising in Clifford analysis. In euclidean n''-space this is :: D=\sum_{j=1}^{n}e_{j}\frac{\partial}{\partial x_{j}} where {''ej: j'' = 1, ..., ''n} is an orthonormal basis for euclidean n-space, and R''n'' is considered to be embedded in a Clifford algebra. This is a special case of the Atiyah-Singer-Dirac operator acting on sections of a spinor bundle. Example 5: For a spin manifold, M'', the Atiyah-Singer-Dirac operator is locally defined as follows: For ''x ∈ M'' and ''e1(x''), ..., ''ej(x'') a local orthonormal basis for the tangent space of ''M at x, the Atiyah-Singer-Dirac operator is :: \sum_{j=1}^{n}e_{j}(x)\tilde{\Gamma}_{e_{j}(x)} , where \tilde{\Gamma} is a lifting of the Levi-Civita connection on M'' to the spinor bundle over ''M. Generalisations In Clifford analysis, the operator D'': ''C∞(R''k'' ⊗ R''n'', S'') → ''C∞(R''k'' ⊗ R''n'', C''k'' ⊗ S'') acting on spinor valued functions defined by : f(x_1,\ldots,x_k)\mapsto \begin{pmatrix} \partial_{\underline{x_1}}f\\ \partial_{\underline{x_2}}f\\ \ldots\\ \partial_{\underline{x_k}}f\\ \end{pmatrix} is sometimes called Dirac operator in ''k Clifford variables. In the notation, S'' is the space of spinors, x_i=(x_{i1},x_{i2},\ldots,x_{in}) are ''n-dimensional variables and \partial_{\underline{x_i}}=\sum_j e_j\cdot \partial_{x_{ij}} is the Dirac operator in the i''-th variable. This is a common generalization of the Dirac operator (''k=1) and the Dolbeault operator (n=2, k'' arbitrary). It is an invariant differential operator, invariant under the action of the group SL(''k) × Spin(n''). The resolution of ''D is known only in some special cases. Υποσημειώσεις Εσωτερική Αρθρογραφία * Διαφορικός Τελεστής * Ολοκληρωτικός Τελεστής * Αριθμητικός Τελεστής * Dirac equation * Clifford algebra * Clifford analysis * Connection * Dolbeault operator * Heat kernel * Spinor bundle Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] Κατηγορία:Μαθηματικοί Τελεστές Κατηγορία:Κβαντική Φυσική